Presentations for three remarkable submonoids of the dihedral inverse monoid on a finite set

In this paper we consider the submonoids \(\mathcal{OPDI}_n\), \(\mathcal{MDI}_n\) and \(\mathcal{ODI}_n\) of the dihedral inverse monoid \(\mathcal{DI}_n\) of all orientation-preserving, monotone and order-preserving transformations, respectively. Our goal is to exhibit presentations for each of these three monoids.